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Math."],"published-print":{"date-parts":[[2020,12]]},"abstract":"Abstract<\/jats:title>We analyze several types of Galerkin approximations of a Gaussian random field$$\\mathscr {Z}:\\mathscr {D}\\times \\varOmega \\rightarrow \\mathbb {R}$$<\/jats:tex-math>Z<\/mml:mi>:<\/mml:mo>D<\/mml:mi>\u00d7<\/mml:mo>\u03a9<\/mml:mi>\u2192<\/mml:mo>R<\/mml:mi><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>indexed by a Euclidean domain$$\\mathscr {D}\\subset \\mathbb {R}^d$$<\/jats:tex-math>D<\/mml:mi>\u2282<\/mml:mo>R<\/mml:mi><\/mml:mrow>d<\/mml:mi><\/mml:msup><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>whose covariance structure is determined by a negative fractional power$$L^{-2\\beta }$$<\/jats:tex-math>L<\/mml:mi>-<\/mml:mo>2<\/mml:mn>\u03b2<\/mml:mi><\/mml:mrow><\/mml:msup><\/mml:math><\/jats:alternatives><\/jats:inline-formula>of a second-order elliptic differential operator$$L:= -\\nabla \\cdot (A\\nabla ) + \\kappa ^2$$<\/jats:tex-math>L<\/mml:mi>:<\/mml:mo>=<\/mml:mo>-<\/mml:mo>\u2207<\/mml:mi>\u00b7<\/mml:mo>(<\/mml:mo>A<\/mml:mi>\u2207<\/mml:mi>)<\/mml:mo><\/mml:mrow>+<\/mml:mo>\u03ba<\/mml:mi>2<\/mml:mn><\/mml:msup><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>. Under minimal assumptions on the domain\u00a0$$\\mathscr {D}$$<\/jats:tex-math>D<\/mml:mi><\/mml:math><\/jats:alternatives><\/jats:inline-formula>, the coefficients$$A:\\mathscr {D}\\rightarrow \\mathbb {R}^{d\\times d}$$<\/jats:tex-math>A<\/mml:mi>:<\/mml:mo>D<\/mml:mi>\u2192<\/mml:mo>R<\/mml:mi><\/mml:mrow>d<\/mml:mi>\u00d7<\/mml:mo>d<\/mml:mi><\/mml:mrow><\/mml:msup><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>,$$\\kappa :\\mathscr {D}\\rightarrow \\mathbb {R}$$<\/jats:tex-math>\u03ba<\/mml:mi>:<\/mml:mo>D<\/mml:mi>\u2192<\/mml:mo>R<\/mml:mi><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>, and the fractional exponent$$\\beta >0$$<\/jats:tex-math>\u03b2<\/mml:mi>><\/mml:mo>0<\/mml:mn><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>, we prove convergence in$$L_q(\\varOmega ; H^\\sigma (\\mathscr {D}))$$<\/jats:tex-math>L<\/mml:mi>q<\/mml:mi><\/mml:msub>(<\/mml:mo>\u03a9<\/mml:mi>\u037e<\/mml:mo>H<\/mml:mi>\u03c3<\/mml:mi><\/mml:msup>(<\/mml:mo>D<\/mml:mi>)<\/mml:mo><\/mml:mrow>)<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>and in$$L_q(\\varOmega ; C^\\delta (\\overline{\\mathscr {D}}))$$<\/jats:tex-math>L<\/mml:mi>q<\/mml:mi><\/mml:msub>(<\/mml:mo>\u03a9<\/mml:mi>\u037e<\/mml:mo>C<\/mml:mi>\u03b4<\/mml:mi><\/mml:msup>(<\/mml:mo>D<\/mml:mi>\u00af<\/mml:mo><\/mml:mover>)<\/mml:mo><\/mml:mrow>)<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>at (essentially) optimal rates for (1) spectral Galerkin methods and (2) finite element approximations. Specifically, our analysis is solely based on$$H^{1+\\alpha }(\\mathscr {D})$$<\/jats:tex-math>H<\/mml:mi>1<\/mml:mn>+<\/mml:mo>\u03b1<\/mml:mi><\/mml:mrow><\/mml:msup>(<\/mml:mo>D<\/mml:mi>)<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>-regularity of the differential operatorL<\/jats:italic>, where$$0<\\alpha \\le 1$$<\/jats:tex-math>0<\/mml:mn><<\/mml:mo>\u03b1<\/mml:mi>\u2264<\/mml:mo>1<\/mml:mn><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>. For this setting, we furthermore provide rigorous estimates for the error in the covariance function of these approximations in$$L_{\\infty }(\\mathscr {D}\\times \\mathscr {D})$$<\/jats:tex-math>L<\/mml:mi>\u221e<\/mml:mi><\/mml:msub>(<\/mml:mo>D<\/mml:mi>\u00d7<\/mml:mo>D<\/mml:mi>)<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>and in the mixed Sobolev space$$H^{\\sigma ,\\sigma }(\\mathscr {D}\\times \\mathscr {D})$$<\/jats:tex-math>H<\/mml:mi>\u03c3<\/mml:mi>,<\/mml:mo>\u03c3<\/mml:mi><\/mml:mrow><\/mml:msup>(<\/mml:mo>D<\/mml:mi>\u00d7<\/mml:mo>D<\/mml:mi>)<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>, showing convergence which is more than twice as fast compared to the corresponding$$L_q(\\varOmega ; H^\\sigma (\\mathscr {D}))$$<\/jats:tex-math>L<\/mml:mi>q<\/mml:mi><\/mml:msub>(<\/mml:mo>\u03a9<\/mml:mi>\u037e<\/mml:mo>H<\/mml:mi>\u03c3<\/mml:mi><\/mml:msup>(<\/mml:mo>D<\/mml:mi>)<\/mml:mo><\/mml:mrow>)<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>-rate. We perform several numerical experiments which validate our theoretical results for (a) the original Whittle\u2013Mat\u00e9rn class, where$$A\\equiv \\mathrm {Id}_{\\mathbb {R}^d}$$<\/jats:tex-math>A<\/mml:mi>\u2261<\/mml:mo>Id<\/mml:mi>R<\/mml:mi><\/mml:mrow>d<\/mml:mi><\/mml:msup><\/mml:msub><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>and$$\\kappa \\equiv {\\text {const.}}$$<\/jats:tex-math>\u03ba<\/mml:mi>\u2261<\/mml:mo>const.<\/mml:mtext><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>, and (b)\u00a0an example of anisotropic, non-stationary Gaussian random fields in$$d=2$$<\/jats:tex-math>d<\/mml:mi>=<\/mml:mo>2<\/mml:mn><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>dimensions, where$$A:\\mathscr {D}\\rightarrow \\mathbb {R}^{2\\times 2}$$<\/jats:tex-math>A<\/mml:mi>:<\/mml:mo>D<\/mml:mi>\u2192<\/mml:mo>R<\/mml:mi><\/mml:mrow>2<\/mml:mn>\u00d7<\/mml:mo>2<\/mml:mn><\/mml:mrow><\/mml:msup><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>and$$\\kappa :\\mathscr {D}\\rightarrow \\mathbb {R}$$<\/jats:tex-math>\u03ba<\/mml:mi>:<\/mml:mo>D<\/mml:mi>\u2192<\/mml:mo>R<\/mml:mi><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>are spatially varying.<\/jats:p>","DOI":"10.1007\/s00211-020-01151-x","type":"journal-article","created":{"date-parts":[[2020,11,16]],"date-time":"2020-11-16T17:04:54Z","timestamp":1605546294000},"page":"819-873","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":19,"title":["Regularity and convergence analysis in Sobolev and H\u00f6lder spaces for generalized Whittle\u2013Mat\u00e9rn fields"],"prefix":"10.1007","volume":"146","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9417-1542","authenticated-orcid":false,"given":"Sonja G.","family":"Cox","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3609-9431","authenticated-orcid":false,"given":"Kristin","family":"Kirchner","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2020,11,16]]},"reference":[{"key":"1151_CR1","unstructured":"Andreev, R.: PPFEM\u2014MATLAB routines for the FEM with piecewise polynomial splines on product meshes (2016). https:\/\/bitbucket.org\/numpde\/ppfem\/. 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